In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

According to this convention, when an index variable appears twice in a single term, once in an upper and once in a lower position, it implies that we are summing over all of its possible values. In typical applications, these are 1,2,3 (for calculations in Euclidean space), or 0,1,2,3 or 1,2,3,4 (for calculations in Minkowski space), but they can have any range, even (in some applications) an infinite set. Abstract index notation is an improvement of Einstein notation. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

In general relativity, the Greek alphabet and the Roman alphabet are used to distinguish whether summing over 1,2,3 or 0,1,2,3 (usually Roman, i, j, ... for 1,2,3 and Greek, μ, ν, ... for 0,1,2,3). As in sign conventions, the convention used in practice varies: Roman and Greek may be reversed. Two-dimensional visualization of space-time distortion. ... The Greek language is written in the Greek alphabet, developed in classical times (ca 9th century B.C.) and passed down to the present. ... ...

Sometimes (as in general relativity), the index is required to appear once as a superscript and once as a subscript; in other applications, all indices are subscripts. See Dual vector space and Tensor product. Two-dimensional visualization of space-time distortion. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

It is important to keep in mind that no new physical laws or ideas result from using Einstein notation; rather, it merely helps in identifying relationships and symmetries often 'hidden' by more conventional notation.

Introduction

In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal unit vectors i, j and k.

If the basis vectors i, j, and k are instead expressed as e₁, e₂, and e₃, a vector can be expressed in terms of a summation:

The innovation of the Einstein notation is the recognition that an index that is repeated twice in an equation implies a summation, and the summation symbol need not be included.

The usefulness of the Einstein notation becomes apparent in the algebraic manipulation of vector equations. For example,

or equivalently:

where

and is the Kronecker delta, which is equal to 1 when i = j, and 0 otherwise. It logically follows that this allows one j in the equation to be converted to an i, or one i to be converted to a j. Then, In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

For the cross product, In mathematics, the cross product is a binary operation on vectors in vector space. ...

where and is the Levi-Civita symbol defined by: In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol, is defined as follows: It is named after Tullio Levi-Civita. ...

which recovers

from

.

Additionally, if , then and . This also highlights that when an index appears once on both sides of the equation, this implies a system of equations instead of a summation:

Formal definitions

In the traditional usage, one has in mind a vector space V with finite dimension n, and a specific basis of V. We can write the basis vectors as e₁, e₂, ..., e_n. Then if v is a vector in V, it has coordinates v₁, ..., v_n relative to this basis. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

The basic rule is:

v = v_i e_i.

In this expression, it is assumed that the term on the right side is to be summed as i goes from 1 to n, because the index i appears on both sides. In that case, the equation is indeed true.

The i is known as a dummy index since the result is not dependent on it; thus we could also write, for example:

v = v_j e_j.

An index that is not summed over is a free index and should be found in each term of the equation or formula.

In contexts where the index must appear once as a subscript and once as a superscript, the basis vectors e_i retain subscripts but the coordinates become vⁱ with superscripts. Then the basic rule is:

v = vⁱ e_i.

The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, , the tensor product of V with itself, has a basis consisting of tensors of the form . Any tensor T in can be written as: In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

.

V*, the dual of V, has a basis e¹, e², ..., eⁿ which obeys the rule

.

Here δ is the Kronecker delta, so is 1 if i=j and 0 otherwise. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

We have also used a superscript for the dual basis, which fits in with a convention requiring summed indices to appear once as a subscript and once as a superscript. In this case, if L is an element in V*, then:

L = L_i eⁱ.

If instead every index is required to be a subscript, then a different letter must be used for the dual basis, say d_i := eⁱ.

The real purpose of the Einstein notation is for formulas and equations that make no mention of the chosen basis. For example, if L and v are as above, then

L(v) = L_i vⁱ,

and this is true for every basis. The next few sections contain further examples of such equations.

Elementary vector algebra and matrix algebra

If V be Euclidean n-space Rⁿ, then there is a standard basis for V, in which e_i is (0,...,0,1,0,...,0), with the 1 in the ith position. Then n-by-n matrices can be thought of as elements of . We can also think of vectors in V as column vectors, or n-by-1 matrices; elements of V* are row vectors, or 1-by-n matrices. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... For the square matrix section, see square matrix. ... In linear algebra, a column vector is an m x 1 matrix, a matrix consisting of a single column. ... In linear algebra, a row vector is a 1 x n matrix, a matrix consisting of a single row. ...

In these examples, all indices will appear as superscripts. (Ultimately, this is because V has an inner product and the chosen basis is orthonormal, as explained in the next section.) In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

If H is a matrix and v is a column vector, then Hv is another column vector. To define w := Hv, we can write:

w_i := H_ij v_j.

Notice that the free index i appears once in every term, while the dummy index j appears twice in a single term.

The distributive law, that H(u + v) = Hu + Hv, can be written: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

H_ij (u_j + v_j) = H_ij u_j + H_ij v_j.

This example also indicates the proof of the distributive law, since the index equation makes direct reference only to certain real numbers, and its validity follows directly from the distributive law for real numbers. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

The transpose of a column vector is a row vector with the same components, and the transpose of a matrix is another matrix whose components are given by swapping the indices. Suppose that we're interested in the product of v^T and H^T. If w (a row vector) is this product, then:

w_i = v_i H_ji.

Thus to say that taking the transpose of a product switches the order of multiplication, we can write:

H_ji v_i = v_i H_ji.

Again, this is obviously true, by the commutative law for real numbers. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

The dot product of two vectors u and v can be written In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this...

.

If n = 3, then we can also write the cross product, using the Levi-Civita symbol. Specifically, if w is u×v, then: In mathematics, the cross product is a binary operation on vectors in vector space. ... In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol, is defined as follows: It is named after Tullio Levi-Civita. ...

.

Here, the Levi-Civita symbol satisfies is 1 if (i,j,k) is an even permutation of (1,2,3), -1 if it's a odd permutation, and 0 if it's not a permutation of (1,2,3) at all. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

You may have noticed in these examples that we often introduced a vector w that would normally not have to be given a specific name using coordinate-free notation. This vector doesn't need to be given a specific name using only index notation either, but the translation between the notations is easier to describe by giving it a name.

With no implicit inner product

If you review the above examples, you'll find that all of them through the distributive law make sense if a summed index must appear once as a subscript and once as a superscript. But the examples from the transpose on don't make sense in that case. This is because they implicitly use the standard inner product on Euclidean space, while the earlier examples do not. In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

In some applications, there is no inner product on V. In these cases, requiring a summed index to appear once as a subscript and once as a superscript can help one avoid errors in calculation, in much the same way as dimensional analysis does. Perhaps more significantly, the inner product may be a primary object of study that shouldn't be suppressed in the notation; this is the case, for example, in general relativity. Then the difference between a subscript and a superscript can be quite significant. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... Two-dimensional visualization of space-time distortion. ...

When an inner product is explicitly referred to, its components are often referred to as g_ij. Note that g_ij = g_ji. Then the formula for the dot product becomes:

.

We can also lower the index on uⁱ by defining

.

Then we have:

.

Note that we have implicitly used g_ij = g_ji here.

Similarly, we can raise an index using the corresponding inner product on V*. The coordinates of this inner product are g^ij, which is (as a matrix) the inverse of g_ij. If you raise an index and then lower it (or the other way around), then you get back where you started. If you raise the i in g_ij, then you get (the Kronecker delta), and if you raise the j in , then you get g^ij. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

If the chosen basis of V is orthonormal, then g_ij = δ_ij and u_i = uⁱ. In this case, the formula for the dot product from the previous section may be recovered. However, if the basis is not orthonormal, then this will not be true; thus, when working with an arbitrary basis one must refer to g_ij explicitly. Furthermore, if the inner product is not positive-definite (as is the case, for example, in special relativity), then even if the basis is chosen to be orthonormal; in this case, g_ij may be +1 or -1 when i=j. Thus, raising and lowering indices are important operations in these applications. In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ...

Of Note

In some fields, Einstein notation is referred to simply as index notation. Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ...

See also

• Bra-ket notation